A Mean-Field Game Model of Electricity Market Dynamics

Abstract

We develop a model for the long-term dynamics of electricity market, based on mean-field games of optimal stopping. Our paper extends the recent contribution (Aïd et al., J. Dyn. Games 8(4):331, 2021) in several ways, making the model much more realistic, especially for describing the medium-term impacts of energy transition on electricity markets. In particular, we allow for an arbitrary number of technologies with endogenous fuel prices, introduce plant construction time and enable the agents to both invest and divest. This makes it possible to describe the role of gas generation as a medium-term substitute for coal, to be replaced by renewable generation in the long term, and enables us to model the events like the 2022 energy price crisis.

Appendix: Proof of the Main Results

We follow the structure of proofs of [2] with some important changes, owing to the facts that agents can both invest and divest and that the fuel prices are endogenously determined.

1.1 Preliminary Lemmas

All results in this section are shown under the assumptions of Theorem 1. The first lemma establishes the compactness of the set $\mathcal R_i$. To this end, we associate the measure flow $(m^i_t)_{0\leq t\leq T}$ with a finite positive measure on $[0,T]\times \mathcal A\times \overline O_i$ defined by $m^i_t\, dt$, and we endow the set $\mathcal V_i$ with the topology of weak convergence of the associated measures. Similarly, the measure flow $(\hat m^i_t)_{0\leq t\leq T}$ is associated with a finite positive measure on $[0,T]\times \overline O_i$ defined by $\hat m^i_t\, dt$, and we endow the set $\widehat {\mathcal V}_i$ with the topology of weak convergence of the associated measures. The sets $\mathcal M_i$ and $\widehat {\mathcal M}_i$ are endowed with the standard weak convergence topology, and the set $\mathcal R_i(\hat \nu ^i_0,\nu ^i_0)$ is endowed with the product topology.

Lemma 1

Fix$i\in \{1,\dots ,N+\overline N\}$and assume that the initial measures$\hat \nu ^i_0$and$\nu ^i_0$satisfy the assumptions (23). Then, the set$\mathcal R_i(\hat \nu ^i_0,\nu ^i_0)$is compact.

Proof

The proof is similar to that of Theorem 2.13 in [14]. Since $\mathcal R_i$ is a subset of a metrizable space, it is enough to show sequential compactness. Consider then a sequence $(\hat \mu ^{i,n}, \hat m^{i,n}, \mu ^{i,n}, m^{i,n})_{n\geq 1} \subset \mathcal R_i$. First, use the test function $\hat u(t,x) = \int _t^T f(s)ds$, where f is nonnegative bounded continuous. Using this test function in (22) yields

$$\displaystyle \begin{aligned} \int_0^T f(s) ds \int_{\overline O_i} \hat m^{i,n}(dx)\leq \int_0^T f(s) ds \int_{\overline O_i} \hat \nu^i_0(dx). {} \end{aligned} $$

(24)

A simple limiting argument (see Lemma 2.8 in [14]) then shows that

$$\displaystyle \begin{aligned} \int_{ \overline O_i} \hat m^{i,n}(dx)\leq \int_{\overline O_i} \hat \nu^i_0(dx) \end{aligned}$$

t-almost everywhere. Now use the test function $\hat u_k(t, x)=(T+1-t)\phi _k(x)$, where

$$\displaystyle \begin{aligned} \phi_k(x)=\ln \left\{1+|x|{}^{3}\left(\frac{3 x^{2}}{5 k^{2}}-\frac{3|x|}{2 k}+1\right)\right\} {\mathbf{1}}_{|x| \leq k}+\ln \left\{1+\frac{k^{3}}{10}\right\} {\mathbf{1}}_{|x|>k}. \end{aligned}$$

For each $k\geq 1$, $\hat u_k\in C^{1, 2}_b([0, T]\times \bar {\mathcal {O}})$ and $\hat u_k$ is non-negative. Using this test function in (22), yields

$$\displaystyle \begin{aligned} &(T+1)\int_{ \overline{\mathcal O}_i} \phi^k(x) \hat\nu^i_0(dx) \\ &\qquad \quad \qquad + \int_{[0,T] \times \overline{\mathcal O}_i} \left\{-\phi^k(x) + (T+1-t)\mathcal L_i \phi^k(x)\right\}\hat m^{i,n}_t(dx)\, dt \geq 0. \end{aligned} $$

Since there exists a constant $C\geq 0$ independent from k such that $\phi _k^{\prime }$ and $\phi ^{\prime \prime }_k$ are bounded by C, together with (24), we obtain that

$$\displaystyle \begin{aligned} \int_{[0,T]\times \mathcal A \times \overline{\mathcal O}_i} \phi^k(x) \hat m^{i,n}_t(dx)\, dt \leq (T+1)\int_{ \overline{\mathcal O}_i} \phi^k(x) \hat\nu^i_0(dx) + C', \end{aligned} $$

for some constant $C'$ which does not depend on n or k. Since, for each x, the sequence $(\phi _k(x))_{k\geq 1}$ is non-decreasing and converges to $\phi _k = \ln (1+|x|{ }^3)$, we conclude by monotone convergence and the assumption of the lemma, that

$$\displaystyle \begin{aligned} \sup_n \int_{[0,T] \times \overline{\mathcal O}_i} \phi(x) \hat m^{i,n}_t(dx)\, dt \leq (T+1)\int_{ \overline{\mathcal O}_i} \phi(x) \hat\nu^i_0(dx) + C'<\infty,{} \end{aligned} $$

(25)

which proves the tightness of the sequence $(\hat m^{i,n})_{n\geq 1}$.

In a similar way, the condition (22) with the same test function yields a bound on $\hat \mu ^{i,n}$:

$$\displaystyle \begin{aligned} \sup_n\int_{[0,T] \times \overline{\mathcal O}_i} \phi(x)\hat \mu^{i,n}(dt,dx)\leq (T+1)\int_{ \overline{\mathcal O}_i} \phi(x) \hat\nu^i_0(dx)+ C'<\infty, \end{aligned} $$

which proves the tightness of this family of measures.

Now, using the test function $u(t,a,x) = \int _t^T f(s) ds$ in the condition (21), and remarking that

$$\displaystyle \begin{aligned} \int_{[0,T]\times \mathcal A \times \overline{\mathcal O}_i} \hat \mu^{i,n}(dt,da,dx) = \int_{\overline{\mathcal O}_i} \hat \nu^{i}_0(dx), \end{aligned}$$

we obtain that

$$\displaystyle \begin{aligned} \int_{\mathcal A \times \overline O_i} m^{i,n}(da,dx)\leq \int_{\overline O_i} \hat \nu^i_0(dx) + \int_{\mathcal A \times \overline O_i} \nu^i_0(da, d x) \end{aligned}$$

t-almost everywhere. Finally, using the test function $u_k(t,a,x) = (T+1-t)\phi _k(x)$ in (21) shows, in a similar way as above, that

$$\displaystyle \begin{aligned} & \sup_n \int_{[0,T]\times \mathcal A \times \overline{\mathcal O}_i} \phi(x) m^{i,n}_t(da,dx)\, dt \leq (T+1)\int_{ \overline{\mathcal O}_i} \phi(x) \hat\nu^i_0(dx) \\&\qquad \qquad +(T+1)\int_{\mathcal A \times \overline O_i} \phi(x)\nu^i_0(da, d x)+ C^{\prime\prime}<\infty,\\ &\sup_n\int_{[0,T]\times \mathcal A \times \overline{\mathcal O}_i} \phi(x) \mu^{i,n}(dt,da,dx)\leq (T+1)\int_{ \overline{\mathcal O}_i} \phi(x) \hat\nu^i_0(dx)\\ &\qquad \qquad +(T+1)\int_{ \mathcal A\times \overline{\mathcal O}_i} \phi(x) \nu^i_0(da,dx)+ C^{\prime\prime}<\infty, \end{aligned} $$

which proves the tightness of the corresponding sequences. Together, all our estimates prove that the sequence $(\hat \mu ^{i,n}, \hat m^{i,n}, \mu ^{i,n}, m^{i,n})_{n\geq 1}$ has a subsequence which converges to a quadruple $(\hat \mu ^{i}, \hat m^{i}, \mu ^{i}, m^{i})$. Since by construction the set $\mathcal R_i(\hat \nu ^i_0,\nu ^i_0)$ is closed with respect to weak convergence, $(\hat \mu ^{i}, \hat m^{i}, \mu ^{i}, m^{i})\subset \mathcal R_i$ and the proof is complete. □

We now prove a technical result regarding a bounded variation property of some specific functionals, which will be used throughout the proofs.

Lemma 2 (A Bounded Variation Property)

Fix$i=N+1, \ldots , N+\bar {N}$. Let$h \in C_b^{1,1,2}([0,T] \times \mathcal {A} \times \overline {\mathcal {O}}_{i})$, $\hat h \in C_b^{1,2}([0,T] \times \overline {\mathcal {O}}_{i})$and$(m^{i},\hat {m}^{i},\mu ^{i},\hat {\mu }^{i}) \in \mathcal {R}_i$. Then, for every$\psi \in C^{1}([0,T]),$

$$\displaystyle \begin{aligned} \int_0^T \psi^\prime(t)\left(\int_{\mathcal{A} \times \overline{\mathcal{O}}_{i}}h(t,a,x)m^{i}_t(da,dx)\right)dt \leq C|\!|\psi|\!|{}_{\infty}. \end{aligned} $$

and

$$\displaystyle \begin{aligned} \int_0^T \psi^\prime(t)\left(\int_{\overline{\mathcal{O}}_{i}}h(t,x)\hat{m}^{i}_t(ddx)\right)dt \leq C|\!|\psi|\!|{}_{\infty}. \end{aligned} $$

For$i=1,\ldots , N$, a similar property holds with functions h which do not depend on x.

Proof

Fix $i=N+1,\ldots , N+\bar {N}$. Consider the test functions $u(t,a,x)=-\psi (t)h(t,a,x)$ and $\hat u(t,x)=-\psi (t)h(t,x)$. Using again the constraints (21)–(22), the boundedness of h and its derivatives, the boundedness of the coefficients of the diffusion processs S, together with the boundedness of the flow of measures $m^{i}$ and $\hat {m}^{i}$ proved in the previous lemma, the result follows. □

The following lemma establishes the existence of measures maximizing the gain functions for fixed price trajectories.

Lemma 3

Fix$i\in \{1,\dots ,N+\overline N\}$and assume that the initial measures$\hat \nu ^i_0$and$\nu ^i_0$satisfy the assumptions (23). Assume that the peak demand$D^p$, the off-peak demand$D^{op}$, the carbon price$P^C$trajectories, the electricity price trajectories$P^p$and$P^{op}$and the fuel price trajectories$P^1,\dots ,P^K$are fixed and have finite variation on$[0,T]$. Then, for each i, there exist a quadruple

$$\displaystyle \begin{aligned} (\hat \mu^i, (\hat m^i_t)_{0\leq t\leq T},\mu^i, ( m^i_t)_{0\leq t\leq T}) \in \mathcal R_i(\hat \nu^i_0,\nu^i_0), \end{aligned}$$

which maxmizes the functional (19) (if$i\in \{1,\dots ,N\}$or (20) (if$i\in \{N+1,\dots ,N+\overline N\}$.

Proof

The proof uses the Lipschitz property of the gain function $G_i$ and the approximation of the price processes with uniformly bounded continuous mappings in $L^1([0,T])$. Since it is very similar to the proof of Lemma 3.2 in [2], the details are omitted here to save space. □

The following lemma establishes the properties of the price process.

Lemma 4

• i. Let

  $$\displaystyle \begin{aligned} (\hat \mu^i, (\hat m^i_t)_{0\leq t\leq T}, \mu^i, ( m^i_t)_{0\leq t\leq T}) \in \mathcal R_i,\quad i=1,\dots,N+\overline N, \end{aligned}$$

  and assume that the peak demand$D^p$, the off-peak demand$D^{op}$and the carbon price$P^C$trajectories have bounded variation on$[0,T]$. Then, the electricity price trajectories$P^p$and$P^{op}$and the fuel price trajectories$P^1,\dots ,P^K$trajectories have bounded variation on$[0,T]$as well.

• ii. Let

  $$\displaystyle \begin{aligned} \left((\hat{\mu}_n^1,\hat{m}_n^1,\mu_n^1,m_n^1), \ldots, (\hat{\mu}^{N+\overline N}_n,\hat{m}^{N+\overline N}_n,\mu^{N+\overline N}_n,m^{N+\overline N}_n)\right) \in \mathcal{R}_1 \times \ldots \times \mathcal{R}_{N+\bar{N}}, \end{aligned}$$

  be a sequence converging to

  $$\displaystyle \begin{aligned} &\left((\hat{\mu}^{1,\star},\hat{m}^{1,\star},\mu^{1,\star},m^{1,\star}), \ldots, (\hat{\mu}^{N+\overline N,\star},\hat{m}^{N+\overline N,\star},\mu^{N+\overline N,\star},m^{N+\overline N,\star})\right) \\ &\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \in \mathcal{R}_1 \times \ldots \times \mathcal{R}_{N+\bar{N}}. \end{aligned} $$

  Assume that the peak demand$D^p$, the off-peak demand$D^{op}$and the carbon price$P^C$trajectories have bounded variation on$[0,T]$. Then there exists a subsequence$(n_l)$such that$P_t^{p,n_l}$, $P_t^{op,n_l}$and$P_t^{k,n_l}$converge in$L^1([0,T])$to$P_t^{p,\star }$, $P_t^{op,\star }$and$P_t^{k,\star }$.

Proof

The proof follows that of Lemma 3.3 in [2] with some important changes and additions due to the fact that the price vector is now multidimensional (it includes not only the electricity price but also the endogenous fuel prices). We shall provide details of the changes and refer to [2] for the parts of the proof, which are very similar to that reference.

Part i

The first step is to show that the peak residual demand $((D^p_t - R_t)^+)_{0\leq t\leq T}$ and the off-peak residual demand $((D^{op}_t - R_t)^+)_{0\leq t\leq T}$ have bounded variation on $[0,T]$. This is done similarly to the first step of [2] using Lemma 2.

In the second step we prove that the fuel prices $P^1,\dots ,P^K$ are bounded. To this end, observe that the function $G_t$ defined in the proof of Proposition 1, equals zero when all prices are zero, and admits the following lower bound:

$$\displaystyle \begin{aligned} G_t (P^p, P^{op},P^1,\dots,P^K) \geq -P^*(c_p D^p_t + c_{op}D^{op}_t) + \sum_{k=1}^K \overline\Phi_k(P^k). \end{aligned} $$

Since the functions $\overline \Phi _k$ for $k=1,\dots ,K$ are positive and strongly convex, it follows that $P^1_t,\dots ,P^K_t$ admit an upper bound $\overline P$ (which may depend on the demand trajectories but they are fixed in this proof). Without loss of generality (by increasing the constant $\overline P$ if needed), we assume that the trajectories of $|P^p - f_i (P^k + e_k P^C)|$ and $|P^{op}- f_i (P^k + e_k P^C)|$ are also bounded from above by $\overline P$ for each $k=1,\dots ,K$.

In the third step we introduce the “discretized” offer functions. Namely for a fixed n, define

$$\displaystyle \begin{aligned} F^{i,n}_{q}(t) = \int_{\mathcal A\times \overline{\mathcal O}_i} m^i_t(da,dx) \lambda_i(a) F_i(q\overline P/n - x) \end{aligned}$$

for $q=-n,\dots ,n$. We can then show, by following the arguments in step 2 of [2], that these functions have bounded variation on $[0,T]$, uniformly on $n,q$, in the sense that for every n and every family of $C^1$ functions $\psi _{q}:[0,T]\to \mathbb R$, $q=-n,\dots ,n$, we have

$$\displaystyle \begin{aligned} \sum_{q=-n}^n \int_0^T F^{i,n}_{q}(t)\psi^{\prime}_{q}(t) dt \leq C \max_{0\leq t\leq T} \sum_{q=-n}^n |\psi_{q}(t)|, \end{aligned}$$

where the constant C does not depend on $q,n$.

In the fourth step we introduce the mollified offer functions. Let $\rho$ be a mollifier supported on $[-1,1]$, set $\rho _m(x) = m\rho (mx)$ and define

$$\displaystyle \begin{aligned} F^{i,n,m}_{q}(t):= \rho_m \star F^{i,n}_{q}(t), \end{aligned}$$

where $F^{i,n}_{q}$ is extended by zero value outside the interval $[0,T]$. Then we can show, by following the arguments in step 3 of [2] that

$$\displaystyle \begin{aligned} \int_0^T \max_{-n\leq q\leq n} \Big|\frac{d}{dt}F^{i,n,m}_{q}(t) \Big|dt \leq C, \end{aligned}$$

for a different constant C.

In the final step, we introduce sequences of mappings $\Theta ^p_n,\Theta ^{op}_n,\Theta ^1_n,\dots ,\Theta ^K_n$, $n\geq 1$, such that $\Theta ^p_n$ and $\Theta ^{op}_n$ approximate the peak and off-peak electricity price, and $\Theta ^1_n,\dots ,\Theta ^K_n$ approximate the fuel prices in the following way: the peak price is approximated by the expression

$$\displaystyle \begin{aligned} P^{p,n}_t := \Theta^p_n((D^p_t-R_t)^+ , (D^{op}_t-R_t)^+, P^C_t, (F^{i,n}_{q}(t))^{i=1,\dots,N}_{q = -n,\dots,n}),{} \end{aligned} $$

(26)

and similarly for the other prices. Lemma 5 shows that one may indeed construct such mappings with required properties. To complete the proof we then follow the arguments in step 4 of [2].

Part ii

This part follows closely the proof of Part ii. of Lemma 3.3 in [2].

□

Lemma 5

There exist sequences of mappings (the arguments in parentheses are the same for all mappings but we only spell them out for the first one):

$$\displaystyle \begin{aligned} \Theta^p_n(u,v,w,(\xi^i_{q})^{i=1,\dots,N}_{q=-n,\dots,n}),\Theta^{op}_n,\Theta^1_n,\dots,\Theta^K_n,\ n\geq 1 \end{aligned}$$

with the following properties:

• 1. $\Theta ^p_n\leq P^*,\ \Theta ^{op}_n\leq P^*,\ \Theta ^1_n\leq \overline P,\dots ,\Theta ^K_n\leq \overline P$.

• 2. There exists a constant C such that$|P^{p,n}_t - P^p_t|\leq \frac {C}{n}$, $|P^{op,n}_t - P^{op}_t|\leq \frac {C}{n}$, $|P^{k,n}_t - P^k_t|\leq \frac {C}{n}$for$k=1,\dots ,K$, where the approximate price$P^{p,n}_t$is defined in (26) and the other approximate prices are defined similarly.

• 3. $\Theta ^p_n,\Theta ^{op}_n,\Theta ^1_n,\dots ,\Theta ^K_n$are differentiable, and their derivatives satisfy, on any compact set,

  $$\displaystyle \begin{aligned} &\Big|\frac{\partial \Theta^\star_n}{\partial u}\Big| \leq C , \quad \Big|\frac{\partial \Theta^\star_n}{\partial v}\Big| \leq C,\quad \Big|\frac{\partial \Theta^\star_n}{\partial w}\Big| \leq C \\ &\mathit{\text{and}} \sum_{i=1,\dots,N, q = -n,\dots, n} \Big|\frac{\partial \Theta^\star_n}{\partial \xi^i_{q}}\Big| \leq C, \end{aligned} $$

  where C is a constant which does not depend on n, where$\star$stands for p, op or$1,\dots ,K$.

Proof

To define these mappings, we consider the linear interpolation maps

$$\displaystyle \begin{aligned} \mathcal{L}_n(y,\xi_{-n},\dots,\xi_n) = ((q_y+1) \overline P/n - y)\xi_{q_y} + (y-q_y \overline P/n)\xi_{q_y + 1}, \end{aligned}$$

for $y\in (-\overline P, \overline P)$, where $q_y = \max \{q:y\overline P/n<y\}$. From these interpolation mappings we define

$$\displaystyle \begin{aligned} F^{k,n}(x,y,(\xi^i_q)^{i=1,\dots,N}_{q=-n,\dots,n})&= \sum_{i: k(i) = k} \mathcal L_n(x- f_i y, \xi^i_{-n},\dots,\xi^i_n)\\ \Psi^{k,n}(x,y,(\xi^i_q)^{i=1,\dots,N}_{q=-n,\dots,n})&= \sum_{i: k(i) = k} f_i \mathcal{L}_n(x- f_i y, \xi^{i}_{-n},\dots,\xi^{i}_{n}). \end{aligned} $$

Let $\overline \Theta ^p_n,\overline \Theta ^{op}_n,\overline \Theta ^1_n,\dots ,\overline \Theta ^K_n$, be the solution to

$$\displaystyle \begin{aligned} c_p \Psi^{k,n}(\theta^p, \theta^k)+c_{op} \Psi^{k,n}(\theta^{op}, \theta^k) = \Phi_k (\theta^k - e_k w),\quad k=1,\dots,K,{} \end{aligned} $$

(27)

where $\Phi _k$ is extended by zero to negative values, and

$$\displaystyle \begin{aligned} u &= F_0(\theta^p) + \sum_{k=1}^K\widehat{ F}^{k,n}(\theta^p, \theta^k) \ \text{or}\ u > F_0(\theta^p) + \sum_{k=1}^K\widehat{F}^{k,n}(\theta^p, \theta^k) \ \text{and}\ \theta^p = P^*,{} \end{aligned} $$

(28)

$$\displaystyle \begin{aligned} v&= F_0(\theta^{op}) + \sum_{k=1}^K\widehat{F}^{k,n}(\theta^{op}, \theta^k)\ \text{or}\ v > F_0(\theta^{op}) \\ &\quad + \sum_{k=1}^K\widehat{ F}^{k,n}(\theta^{op}, \theta^k)\ \text{and}\ \theta^{op} = P^*,{} \end{aligned} $$

(29)

where we omitted the arguments $\xi ^i_q$ to save space. The value of the mappings $\Theta ^p_n,\Theta ^{op}_n,\Theta ^1_n,\dots ,\Theta ^K_n$ at the point $(u,v,(\xi ^i_{q})^{i=1,\dots ,N}_{q=-n,\dots ,n})$ is defined as follows:

$$\displaystyle \begin{aligned} \Theta^p_n = \overline \Theta^p_n,\quad \Theta^{op}_n = \overline \Theta^{op}_n,\quad \Theta^1_n = \overline\Theta^1_n-e_1 w,\dots,\Theta^K_n = \overline\Theta^K_n-e_K w. \end{aligned}$$

To show that $\Theta ^p_n,\Theta ^{op}_n,\Theta ^1_n,\dots ,\Theta ^K_n$ are uniquely determined, define

$$\displaystyle \begin{aligned} \overline{\mathcal L}_n(y,\xi_{-n},\dots,\xi_n):=\int_0^y \mathcal L_n(y',\xi_{-n},\dots,\xi_n) dy', \end{aligned}$$

and consider the mapping

$$\displaystyle \begin{aligned} &G^n(\theta^p, \theta^{op}, \theta^1,\dots,\theta^K):=\sum_{k=1}^K \sum_{i:k(i)=k}\{c_p \overline{\mathcal L}_n(\theta^p-f_i \theta^k)+c_{op} \overline{\mathcal L}_n(\theta^{op}-f_i \theta^k)\} \\ & \qquad +c_p G_0(\theta^p) + c_{op}G_0(\theta^{op})- c_p u \theta^p - c_{op} v\theta^{op} + \sum_{k=1}^K \overline \Phi_k(\theta^k), \end{aligned} $$

where we used some notation of Proposition 1. This mapping is strongly convex (due to strong convexity of $G_0$ and $\overline \Phi _1,\dots ,\overline \Phi _K$) and the vector $(\overline \Theta ^p_n,\overline \Theta ^{op}_n,\overline \Theta ^1_n,\dots ,\overline \Theta ^K_n)$ is its unique minimizer.

Next, observe that by construction,

$$\displaystyle \begin{aligned} F^{i,n}_{q_y} \leq \mathcal L_n(y,F^{i,n}_{-n}(t),\dots,F^{i,n}_n(t))\leq F^{i,n}_{q_y+1} \end{aligned}$$

It follows that

$$\displaystyle \begin{aligned} &\int_{\mathcal A\times \overline{\mathcal O}_i} m^i_t(da,dx) \lambda_i(a) F_i(y-\overline P/n - x)\leq L_n(y,F^{i,n}_{-n}(t),\dots,F^{i,n}_n(t))\\ &\qquad \qquad \qquad \qquad \qquad \qquad \leq \int_{\mathcal A\times \overline{\mathcal O}_i} m^i_t(da,dx) \lambda_i(a) F_i(y+\overline P/n - x). \end{aligned} $$

Integrating the upper bound, we get:

$$\displaystyle \begin{aligned} \overline{\mathcal L}_n(y) &\leq \int_{\mathcal A\times \overline{\mathcal O}_i} m^i_t(da,dx) \lambda_i(a) \int_0^y F_i(y'+\overline P/n - x) dy' \\ & \leq \int_{\mathcal A\times \overline{\mathcal O}_i} m^i_t(da,dx) \lambda_i(a) G_i(y-x) + \frac{C}{n}, \end{aligned} $$

for some constant C, by the smoothness of $F_i$ and integrability of $m^i$. Proceeding similarly for the lower bound and summing up the terms, we finally get (with the notation of Proposition 1):

$$\displaystyle \begin{aligned} \Big| \sum_{k=1}^K \sum_{i:k(i)=k}\{c_p \overline{\mathcal L}_n(\theta^p-f_i \theta^k)&+c_{op} \overline{\mathcal L}_n(\theta^{op}-f_i \theta_k)\} \\ &- \sum_{k=1}^K \{c_p G^k_t(\theta^p,\theta^k)-c_{op} G^k_t(\theta^{op},\theta^k)\} \Big|\leq \frac{C}{n}. \end{aligned} $$

Property 2 of the lemma now follows from strong convexity.

We now concentrate on property 3. To save space, we denote simply $\boldsymbol \theta := (\theta ^p,\theta ^{op},\theta ^1,\dots ,\theta ^K)$. Let $\boldsymbol {\Theta }^{\prime }_n$ be the minimizer of $G_n(\boldsymbol \theta )-c_p (u^\prime -u)\boldsymbol {\theta }$. It is characterized by the equation

$$\displaystyle \begin{aligned} \nabla G_n(\boldsymbol{\Theta}^{\prime}_n) = c^p(u'-u){\mathbf{e}}^p, \end{aligned}$$

where ${\mathbf {e}}^p$ is the $K+2$-dimensional vector with 1 in the first position and 0 elsewhere. Differentiating both sides using the smoothness of $G^n$, we get:

$$\displaystyle \begin{aligned} \nabla^2 G^n(\boldsymbol{\Theta}^{\prime}_n)\frac{\partial\boldsymbol{\Theta}^{\prime}_n}{\partial u'} = c^p \boldsymbol e^p. \end{aligned}$$

By strong convexity, $\nabla ^2 G^n(\boldsymbol {\Theta }^{\prime }_n)$ is invertible and bounded from below, whence the boundedness of $\frac {\partial \boldsymbol {\Theta }^{\prime }_n}{\partial u'}$. The boundedness of the derivative with respect to v is obtained in a similar fashion. To analyze the derivative with respect to w, in a similar fashion, we may compute:

$$\displaystyle \begin{aligned} \nabla^2 G^n(\boldsymbol{\Theta}^{\prime}_n)\frac{\partial\boldsymbol{\Theta}^{\prime}_n}{\partial w'} = \mathbf e_k \sum_{k=1}^K\Phi^{\prime}_k (\theta^k-e_k w). \end{aligned}$$

Since the RHS is bounded on compact sets, we conclude that the derivative with respect to w is bounded.

Finally analyze the derivatives with respect to $\xi ^i_q$, to save space, we denote by $\boldsymbol \xi$ the family $(\xi ^i_{q})^{i=1,\dots ,N}_{q=-n,\dots ,n})$, and denote by $\boldsymbol {\Theta }^{\prime }_n$ the minimizer of $G_n$ with $\boldsymbol \xi$ replaced by $\boldsymbol \xi '$. Then, proceding as above and differentiating both sides with respect to $\xi ^{i\prime }_q$, we get:

$$\displaystyle \begin{aligned} &\nabla^2 G_n(\boldsymbol{\Theta}^{\prime}_n)\frac{\partial \Theta^{\prime}_n}{\partial \xi^{i}_q} + ({\mathbf{e}}^p - f_i{\mathbf{e}}^{k(i)})c_p\{ ((q_{y_p}+1) \overline P/n - y_p)\mathbf 1_{q=q_{y_p}} \\ &\qquad \qquad \qquad \qquad \qquad + (y_p-q_{y_p} \overline P/n)\mathbf 1_{q=q_{y_p}+1}\}\\ &+({\mathbf{e}}^{op} - f_i{\mathbf{e}}^{k(i)})c_{op}\{ ((q_{y_{op}}+1) \overline P/n - y_{op})\mathbf 1_{q=q_{y_{op}}} + (y_{op}-q_{y_{op}} \overline P/n)\mathbf 1_{q=q_{y_{op}}+1}\}, \end{aligned} $$

where $y_p = \theta ^{p\prime } - f_i \theta ^{k\prime }$ and $y_{op} = \theta ^{op\prime } - f_i\theta ^{k\prime }$. By strong convexity we obtain the boundedness of $\frac {\partial \Theta ^{\prime }_n}{\partial \xi ^{i}_q}$. In addition, for fixed i, among the derivatives $\frac {\partial \Theta ^{\prime }_n}{\partial \xi ^{i}_q}$, $q=-n,\dots ,n$, only at most four are nonzero (because of the indicator functions). Thus, sum of derivatives is bounded uniformly on n. □

1.2 Proof of Theorem 1

For $i=1,\ldots , N+\bar {N}$, denote by ${\boldsymbol {m}^{i}}$ the family $(\hat {\mu }^{i},(\hat {m}^{i}_t)_{0 \leq t \leq T},\mu ^{i},(m^{i}_t)_{0 \leq t \leq T})$ and ${\boldsymbol {{m}}}=(\boldsymbol {m}^1,\boldsymbol {m}^2,\ldots , \boldsymbol {m}^{N+\bar {N}})$. With this notation, the condition of the lemma simplifies to $\boldsymbol {m}_n\to \boldsymbol {m}^*$. We also define, for $i=1, \ldots , N$,

$$\displaystyle \begin{aligned} \Psi^{i}(\boldsymbol{m}^{i},\boldsymbol{m}) &=\int_{[0,T]\times \mathcal A \times \overline{\mathcal O}_i} m^i_t(da, dx) e^{-\rho t}\lambda_i(a)\\ &\quad \times\left[c_p G_i(P^p_t(\boldsymbol{m}) - e_i P^C_t-f_i P^{k(i)}_t(\boldsymbol{m}) - x )\right. \\ & \left.\quad +c_{op} G_i(P^{op}_t(\boldsymbol{m}) - e_i P^C_t-f_i P^{k(i)}_t(\boldsymbol{m}) - x )-\kappa_i \right]dt \notag\\ &\quad - K_i \int_{[0,T] \times \overline{\mathcal O}_i} \hat \mu^i(dt, dx) e^{-(\rho+\gamma_i) t}\\ &\quad + \widetilde K_i \int_{[0,T]\times \mathcal A \times \overline{\mathcal O}_i} \mu^i(dt, da, dx) e^{-(\rho+\gamma_i) t} \end{aligned} $$

and, for $i=N+1,\ldots ,N+\bar {N}$,

$$\displaystyle \begin{aligned} \bar{\Psi}^{i}(\boldsymbol{m}^i,\boldsymbol{m})&=\int_{[0,T]\times \mathcal A \times \overline{\mathcal O}_i} m^i_t(da, dx) e^{-\rho t}\lambda_i(a)\\ &\quad \times\left[ (c_p P^p_t(\boldsymbol{m})+c_{op}P^{op}_t(\boldsymbol{m})) x - {\kappa_i}\right] dt \\ &\quad - K_i \int_{[0,T] \times \overline{\mathcal O}_i} \hat \mu^i(dt, dx) e^{-(\rho+\gamma_i) t}\\ &\quad + \widetilde K_i \int_{[0,T]\times \mathcal A \times \overline{\mathcal O}_i} \mu^i(dt, da, dx) e^{-(\rho+\gamma_i) t}. \end{aligned} $$

Consider the set valued map

$$\displaystyle \begin{aligned} \Theta: \mathcal{R}(\hat\nu^1_0,\nu^1_0)\times \ldots \times \mathcal{R}(\hat\nu^{N+\bar{N}}_0,\nu^{N+\bar{N}}_0)\to 2^{\mathcal{R}(\hat\nu^1_0,\nu^1_0)\times \ldots \times \mathcal{R}(\hat\nu^{N+\bar{N}}_0,\nu^{N+\bar{N}}_0)},\end{aligned}$$

which is given by

$$\displaystyle \begin{aligned} &\Theta:{\boldsymbol{m}} \mapsto \otimes_{i=1}^{N}\text{argmax}_{\boldsymbol{m}^{i} \in \mathcal R_i(\hat\nu^i_0,\nu^i_0)} \Psi^{i}(\boldsymbol{m}^{i},\boldsymbol{{m}})\\ &\qquad \qquad \qquad \qquad \qquad \otimes_{i=N+1}^{N+\bar N}\text{argmax}_{\boldsymbol{m}^{i} \in \mathcal R_i(\hat\nu^i_0,\nu^i_0)} \bar{\Psi}^{i}(\boldsymbol{m}^{i},\boldsymbol{{m}}). \end{aligned} $$

(30)

To prove the existence of an equilibrium ${\boldsymbol {m}}^\star \in \mathcal {R}(\hat \nu ^1_0,\nu ^1_0)\times \ldots \times \mathcal {R}(\hat \nu ^{N+\bar {N}}_0,\nu ^{N+\bar {N}}_0)$ by using the Fan-Glicksberg fixed point theorem, we only need to show that $\Theta$ has closed graph, which is defined as

$$\displaystyle \begin{aligned} \text{Gr}(\Theta)=\left\{({\boldsymbol{m}},{\boldsymbol{m}}') \in \left(\mathcal{R}(\hat\nu^1_0,\nu^1_0)\times \ldots \times \mathcal{R}(\hat\nu^{N+\bar{N}}_0,\nu^{N+\bar{N}}_0)\right)^2: {\boldsymbol{m}}' \in \Theta({\boldsymbol{m}})\right\}.\end{aligned}$$

To show that $\text{Gr}(\Theta )$ is closed, we have to prove that for any sequence $({\boldsymbol {m}}_{1,n}, {\boldsymbol {m}}_{2,n})$ which weakly converges to $(\widetilde {{\boldsymbol {m}}}_1, \widetilde {{\boldsymbol {m}}}_2) \in \left (\mathcal {R}(\hat \nu ^1_0,\nu ^1_0)\times \ldots \times \mathcal {R}(\hat \nu ^{N+\bar {N}}_0,\nu ^{N+\bar {N}}_0)\right )^2$ such that ${\boldsymbol {m}}_{2,n} \in \Theta ({\boldsymbol {m}}_{1,n})$, we have that $\widetilde {{\boldsymbol {m}}}_2 \in \Theta (\widetilde {{\boldsymbol {m}}}_1)$.

To prove this result, we fix first $i=N+1,\ldots N+\bar {N}$ and show that

$$\displaystyle \begin{aligned} & \lim_{n \rightarrow \infty} \int_{[0,T]\times \mathcal A \times \overline{\mathcal O}_i} m^i_{2,n,t}(da, dx) e^{-\rho t}\lambda_i(a)\\ &\quad \times\left[ (c_p P^p_t({\boldsymbol{m}}_{1,n})+c_{op}P_t^{op}({\boldsymbol{m}}_{1,n})) x - {\kappa_i}\right] dt \\&\quad - K_i \int_{[0,T]\times \overline{\mathcal O}_i} \hat \mu^i_{2,n}(dt, dx) e^{-(\rho+\gamma_i) t}\\ &\quad + \widetilde K_i \int_{[0,T]\times \mathcal A \times \overline{\mathcal O}_i} \mu^i_{2,n}(dt, da, dx) e^{-(\rho+\gamma_i) t} \\ & = \int_{[0,T]\times \mathcal A \times \overline{\mathcal O}_i} \widetilde{m}^i_{2,t}(da, dx) e^{-\rho t}\lambda_i(a)\left[ (c_p P^p_t(\widetilde{{\boldsymbol{m}}}_{1})+c_{op}P_t^{op}(\widetilde{{\boldsymbol{m}}}_{1})) x - {\kappa_i}\right] dt \\&\quad - K_i \int_{[0,T] \times \overline{\mathcal O}_i} \widetilde{\hat \mu}^i_2(dt, dx) e^{-(\rho+\gamma_i) t} + \widetilde K_i\! \int_{[0,T]\times \mathcal A \times \overline{\mathcal O}_i} \widetilde{\mu}^i_2(dt, da, dx) e^{-(\rho+\gamma_i) t}. \end{aligned} $$

Using Lemma 2, we can prove that the total variation of the map $t \mapsto h(t)$, with $h(t)$ given by $h(t)=\int _{ \bar {\mathcal {O}}_i \times \mathcal {A}} x \lambda _i(a)m^i_{2,n,t}(da,dx)$ is uniformly bounded with respect to n, and therefore by Theorem 3.23 in [5], we get that there exists a subsequence $(n_k)$ such that the maps $\int _{\mathcal {A} \times \bar {\mathcal {O}}_i} x \lambda _i(a)m^i_{2,n_k,\cdot }(da,dx)$ converges in $L^1([0,T])$ to some limit which can be identified due to the weak convergence with $\int _{\mathcal {A} \times \bar {\mathcal {O}}_i} x \lambda _i(a)\widetilde {m}^i_{2,\cdot }(da,dx)$. Moreover, in view of Lemma 4 item ii., $P_t^p({\boldsymbol {m}}_{1,n})$ and $P_t^{op}({\boldsymbol {m}}_{1,n})$ converge (up to a subsequence) to $P^{p}_t(\widetilde {{\boldsymbol {m}}}_1)$ and $P^{op}_t(\widetilde {{\boldsymbol {m}}}_2)$. Since both terms are bounded, we derive that that the integral of their product converges too. The convergence of the integrals with respect to $\hat {\mu }^{i}_{2,n}$ and ${\mu }^{i}_{2,n}$ follows by weak convergence of the measures.

It remains to show that, for $i=1, \ldots , N,$

$$\displaystyle \begin{aligned} &\lim_{n \mapsto \infty} \int_{[0,T]\times \mathcal A \times \overline{\mathcal O}_i} m^i_{2,n,t}(da, dx) e^{-\rho t}\lambda_i(a)\\ &\quad \times\left[c_p G_i(P^p_t({\boldsymbol{m}}_{1,n}) - e_i P^C_t-f_i P^{k(i)}_t({\boldsymbol{m}}_{1,n}) - x )\right. \\ & \quad \left.+c_{op} G_i(P^{op}_t({\boldsymbol{m}}_{1,n}) - e_i P^C_t-f_i P^{k(i)}_t({\boldsymbol{m}}_{1,n}) - x )-\kappa_i \right]dt \\ &\quad - K_i \int_{[0,T] \times \overline{\mathcal O}_i} \hat \mu^i_{2,n}(dt, dx) e^{-(\rho+\gamma_i) t} \\ &\quad + \widetilde K_i \int_{[0,T]\times \mathcal A \times \overline{\mathcal O}_i} \mu^i_{2,n}(dt, da, dx) e^{-(\rho+\gamma_i) t} \\ &=\int_{[0,T]\times \mathcal A \times \overline{\mathcal O}_i} \widetilde{m}^i_{2,t}(da, dx) e^{-\rho t}\lambda_i(a)\\ &\quad \times\left[c_p G_i(P^p_t(\widetilde{{\boldsymbol{m}}}_{1}) - e_i P^C_t-f_i P^{k(i)}_t(\widetilde{{\boldsymbol{m}}}_1) - x )\right. \\ & \quad \left.+c_{op} G_i(P^{op}_t(\widetilde{{\boldsymbol{m}}}_1) - e_i P^C_t-f_i P^{k(i)}_t(\widetilde{{\boldsymbol{m}}}_1) - x )-\kappa_i \right]dt \\ &- K_i \int_{[0,T] \times \overline{\mathcal O}_i} \widetilde{\hat \mu}^{i}_2(dt, dx) e^{-(\rho+\gamma_i) t} + \widetilde K_i \int_{[0,T]\times \mathcal A \times \overline{\mathcal O}_i} \widetilde{\mu}^{i}_2(dt, da, dx) e^{-(\rho+\gamma_i) t}. \end{aligned} $$

Recall that, for each $i=1,\ldots ,N$, $G_{i}$ is a Lipschitz function with constant 1. We then obtain

$$\displaystyle \begin{aligned} &\left|\int_{[0,T]\times \mathcal A \times \overline{\mathcal O}_i} m^i_{2,n,t}(da, dx) e^{-\rho t}\lambda_i(a)\right.\\ &\times \left[c_p G_i(P^p_t({\boldsymbol{m}}_{1,n}) - e_i P^C_t-f_i P^{k(i)}_t({\boldsymbol{m}}_{1,n}) - x ) \right. \\ & \left.+c_{op} G_i(P^{op}_t({\boldsymbol{m}}_{1,n}) - e_i P^C_t-f_i P^{k(i)}_t({\boldsymbol{m}}_{1,n}) - x )-\kappa_i \right]dt \\ & - K_i \int_{[0,T]\times \overline{\mathcal O}_i} \hat \mu^i_{2,n}(dt, dx) e^{-(\rho+\gamma_i) t} + \widetilde K_i \int_{[0,T]\times \mathcal A \times \overline{\mathcal O}_i} \mu^i_{2,n}(dt, da, dx) e^{-(\rho+\gamma_i) t} \\ & - \int_{[0,T]\times \mathcal A \times \overline{\mathcal O}_i} \widetilde{m}^i_{2,t}(da, dx) e^{-\rho t}\lambda_i(a)\\ &\times\left[c_p G_i(P^p_t(\widetilde{{\boldsymbol{m}}}_{1}) - e_i P^C_t-f_i P^{k(i)}_t(\widetilde{{\boldsymbol{m}}}_1) - x )\right. \\ & \left. +c_{op} G_i(P^{op}_t(\widetilde{{\boldsymbol{m}}}_1) - e_i P^C_t-f_i P^{k(i)}_t(\widetilde{{\boldsymbol{m}}}_1) - x )-\kappa_i \right]dt \\ & \left. - K_i \int_{[0,T] \times \overline{\mathcal O}_i} \widetilde{\hat \mu}^{i}_2(dt, dx) e^{-(\rho+\gamma_i) t} + \widetilde K_i \int_{[0,T]\times \mathcal A \times \overline{\mathcal O}_i} \widetilde{\mu}^{i}_2(dt, da, dx) e^{-(\rho+\gamma_i) t} \right| \end{aligned} $$

$$\displaystyle \begin{aligned} & \leq \int_{[0,T]\times \mathcal A \times \overline{\mathcal O}_i} m^i_{2,n,t}(da, dx) e^{-\rho t}\lambda_i(a)\Big[c_p |P^p_t({\boldsymbol{m}}_{1,n}) - P^p_t(\widetilde{{\boldsymbol{m}}}_{1})| \\ & +c_{op} |P^{op}_t({\boldsymbol{m}}_{1,n}) - P^{po}_t(\widetilde{{\boldsymbol{m}}}_{1})|+f_i |P^{k(i)}_t({\boldsymbol{m}}_{1,n}) - P^{k(i)}_t({\widetilde{\boldsymbol{m}}}_{1})|\Big] \\ &+\Big|\int_{[0,T]\times \mathcal A \times \overline{\mathcal O}_i} (m^i_{2,n,t}(da, dx)-\widetilde{m}^{i}_{2,t}(da,dx)) e^{-\rho t}\lambda_i(a) \\ &\times \Big[c_p G_i(P^p_t(\widetilde{{\boldsymbol{m}}}_{1}) - e_i P^C_t-f_i P^{k(i)}_t(\widetilde{{\boldsymbol{m}}}_1) - x ) \\ & +c_{op} G_i(P^{op}_t(\widetilde{{\boldsymbol{m}}}_1) - e_i P^C_t-f_i P^{k(i)}_t(\widetilde{{\boldsymbol{m}}}_1) - x )-\kappa_i \Big]dt \Big| \\ &+K_i \Big| \int_{[0,T]\times \overline{\mathcal O}_i} e^{-(\rho+\gamma_i) t}(\hat \mu^i_{2,n}(dt, dx)-\widetilde{\hat \mu}^i_{2}(dt, dx)) \Big| \\ &+ \widetilde K_i \Big|\int_{[0,T]\times \mathcal A \times \overline{\mathcal O}_i} e^{-(\rho+\gamma_i) t} (\mu^i_{2,n}(dt, da, dx)-\widetilde{\mu}^i_{2}(dt, da, dx))\Big|. \end{aligned} $$

Using similar arguments as in Lemma 2, one can show that the maps $t \mapsto h_n(t)$, with $h_n(t)=\int _{ \mathcal A \times \overline {\mathcal O}_i} m^i_{2,n,t}(da, dx) e^{-\rho t}\lambda _i(a)$ are of bounded variation, uniformly with respect to n, and together with Theorem 3.23 in [5], we obtain that

$$\displaystyle \begin{aligned} {} \int_{ \mathcal A \times \overline{\mathcal O}_i} m^i_{2,n,\cdot}(da, dx) e^{-\rho t}\lambda_i(a) \mapsto \int_{ \mathcal A \times \overline{\mathcal O}_i} \widetilde{m}^i_{2,\cdot}(da, dx) e^{-\rho t}\lambda_i(a) \end{aligned} $$

(31)

in $L^1([0,T])$. Furthermore, using again Lemma 4 item ii. together with the boundedness of the price functionals $P^p$, $P^{op}$ and $P^k$, we derive the convergence to 0 of the first term from the RHS of the above inequality. The convergence of the third and fourth terms to 0 is a consequence of the weak convergence of the sequence of measures ${\boldsymbol {m}}_{1,n}$ towards $\widetilde {{\boldsymbol {m}}}_{1}$. For the second term, we consider a sequence of bounded continuous functions $P_t^{p,m}$ (resp. $P_t^{op,m}$ and $P_t^{k,m}$) approximating in $L^1([0,T])$ the price functionals $P^{p}_t(\widetilde {{\boldsymbol {m}}}_1)$ (resp. $P^{op}_t(\widetilde {{\boldsymbol {m}}}_1)$ and $P^{k,p}_t(\widetilde {{\boldsymbol {m}}}_1)$). Using again the Lipschitz property of the map $G_{i}$, we derive that,

$$\displaystyle \begin{aligned} &\Big|\int_{[0,T]\times \mathcal A \times \overline{\mathcal O}_i} (m^i_{2,n,t}(da, dx)-\widetilde{m}^{i}_{2,t}(da,dx)) e^{-\rho t}\lambda_i(a) \\&\qquad \times \Big[c_p G_i(P^p_t(\widetilde{{\boldsymbol{m}}}_{1}) - e_i P^C_t-f_i P^{k(i)}_t(\widetilde{{\boldsymbol{m}}}_1) - x ) \\ & \qquad +c_{op} G_i(P^{op}_t(\widetilde{{\boldsymbol{m}}}_1) - e_i P^C_t-f_i P^{k(i)}_t(\widetilde{{\boldsymbol{m}}}_1) - x )-\kappa_i \Big]dt \Big| \\ & \leq \int_0^T e^{-\rho t} \Big|\int_{ \mathcal A \times \overline{\mathcal O}_i} (m^i_{2,n,t}(da, dx)-\widetilde{m}^i_{2,t}(da, dx))\Big| \lambda_i(a)\\&\qquad \times\!\Big[\!c_p |P^p_t(\widetilde{{\boldsymbol{m}}_{1}}) - P^{p,m}_t|+c_{op} |P^{op}_t(\widetilde{{\boldsymbol{m}}_{1}}) - P^{op,m}_t| +f_i |P^{k(i)}_t(\widetilde{{\boldsymbol{m}}_{1}}) - P^{k(i),m}_t|\Big] \\ & \qquad +\Big|\int_{[0,T]\times \mathcal A \times \overline{\mathcal O}_i} (m^i_{2,n,t}(da, dx)-\widetilde{m}^{i}_{2,t}(da,dx)) e^{-\rho t}\lambda_i(a)\\&\qquad \times \Big[c_p G_i(P^{p,m}_t - e_i P^C_t-f_i P^{k(i,m}_t - x ) \\ & \qquad +c_{op} G_i(P^{op,m}_t - e_i P^C_t-f_i P^{k(i),m}_t - x )-\kappa_i \Big]dt \Big|. \end{aligned} $$

The second term in the RHS of the above inequality converges to 0 in view of the weak convergence of measures, while the convergence to 0 of the first term follows by using again (31) and dominated convergence.

1.3 Proof of Proposition 2

Assume that there are two Nash equilibria, corresponding to families of measures and measure flows

$$\displaystyle \begin{aligned} (\hat \mu^i, (\hat m^i_t)_{0\leq t\leq T},\mu^i, ( m^i_t)_{0\leq t\leq T}) \quad \text{and}\quad (\hat {\bar\nu}^i,\hat {\bar\mu}^i, (\hat {\bar m}^i_t)_{0\leq t\leq T}, \bar\nu^i,\bar\mu^i, ( \bar m^i_t)_{0\leq t\leq T}) \end{aligned}$$

for $i=1,\dots ,N+\overline N$ and price vectors

$$\displaystyle \begin{aligned} P^p,P^{op}, P^1,\dots, P^K \quad \text{and}\quad \overline P^p,\overline P^{op}, \overline P^1,\dots, \overline P^K. \end{aligned}$$

By definition of the Nash equilibrium, we have, for conventional agents,

$$\displaystyle \begin{aligned} &\int_{[0,T]\times \mathcal A \times \overline{\mathcal O}_i} m^i_t(da, dx) e^{-\rho t}\lambda_i(a)(c_p G_i(P^p_t - e_i P^C_t-f_i P^{k(i)}_t - x )\\ &+c_{op} G_i(P^{op}_t - e_i P^C_t-f_i P^{k(i)}_t - x )-\kappa_i) dt \\ &- K_i \int_{[0,T] \times \overline{\mathcal O}_i} \hat \mu^i(dt, dx) e^{-(\rho+\gamma_i) t} + \widetilde K_i \int_{[0,T]\times \mathcal A \times \overline{\mathcal O}_i} \mu^i(dt, da, dx) e^{-(\rho+\gamma_i) t}\\ &\geq \int_{[0,T]\times \mathcal A \times \overline{\mathcal O}_i} \bar m^i_t(da, dx) e^{-\rho t}\lambda_i(a)(c_p G_i(P^p_t - e_i P^C_t-f_i P^{k(i)}_t - x )\\ &+c_{op} G_i(P^{op}_t - e_i P^C_t-f_i P^{k(i)}_t - x )-\kappa_i) dt \\ &- K_i \int_{[0,T] \times \overline{\mathcal O}_i} \hat {\bar\mu}^i(dt, dx) e^{-(\rho+\gamma_i) t} + \widetilde K_i \int_{[0,T]\times \mathcal A \times \overline{\mathcal O}_i} \bar\mu^i(dt, da, dx) e^{-(\rho+\gamma_i) t} \end{aligned} $$

and

$$\displaystyle \begin{aligned} &\int_{[0,T]\times \mathcal A \times \overline{\mathcal O}_i} \bar m^i_t(da, dx) e^{-\rho t}\lambda_i(a)(c_p G_i(\overline P^p_t - e_i P^C_t-f_i \overline P^{k(i)}_t - x )\\ &+c_{op} G_i(\overline P^{op}_t - e_i P^C_t-f_i \overline P^{k(i)}_t - x )-\kappa_i) dt \\ &- K_i \int_{[0,T] \times \overline{\mathcal O}_i} \hat{\bar\mu}^i(dt, dx) e^{-(\rho+\gamma_i) t} + \widetilde K_i \int_{[0,T]\times \mathcal A \times \overline{\mathcal O}_i} \bar\mu^i(dt, da, dx) e^{-(\rho+\gamma_i) t}\\ &\geq \int_{[0,T]\times \mathcal A \times \overline{\mathcal O}_i} \bar m^i_t(da, dx) e^{-\rho t}\lambda_i(a)(c_p G_i(\overline P^p_t - e_i P^C_t-f_i \overline P^{k(i)}_t - x )\\ &+c_{op} G_i(\overline P^{op}_t - e_i P^C_t-f_i \overline P^{k(i)}_t - x )-\kappa_i) dt \\ &- K_i \int_{[0,T] \times \overline{\mathcal O}_i} \hat {\bar\mu}^i(dt, dx) e^{-(\rho+\gamma_i) t} + \widetilde K_i \int_{[0,T]\times \mathcal A \times \overline{\mathcal O}_i} \bar\mu^i(dt, da, dx) e^{-(\rho+\gamma_i) t}, \end{aligned} $$

and similarly for the renewable agents. Summing up the two expressions, we get, for conventional agents,

$$\displaystyle \begin{aligned} &\int_{[0,T]\times \mathcal A \times \overline{\mathcal O}_i} (m^i_t-\bar m^i_t)(da,dx) e^{-\rho t}\lambda_i(a)(c_p G_i(P^p_t - e_i P^C_t-f_i P^{k(i)}_t - x )\\ &\ \ +c_{op} G_i(P^{op}_t - e_i P^C_t-f_i P^{k(i)}_t - x )-c_p G_i(\overline P^p_t - e_i P^C_t-f_i \overline P^{k(i)}_t - x )\\ &\qquad \qquad \qquad \qquad \qquad \qquad -c_{op} G_i(\overline P^{op}_t - e_i P^C_t-f_i \overline P^{k(i)}_t - x )) dt\geq 0, \end{aligned} $$

and for renewable agents,

$$\displaystyle \begin{aligned} &\int_{[0,T]\times \mathcal A \times \overline{\mathcal O}_i} (m^i_t-\bar m^i_t)(da, dx) e^{-\rho t}\bar\lambda_i(a)\\ &\quad \times (c_p P^p_t+c_{op}P^{op}_t-c_p \overline P^p_t-c_{op}\overline P^{op}_t) x \,dt\geq 0. \end{aligned} $$

Remark that the terms involving $\mu$ and $\hat \mu$ cancel out in the expressions because they do not depend on the prices. Now, summing up these expressions over all agent types, we get:

$$\displaystyle \begin{aligned} &\int_0^T \{G_t(P^p_t, P^{op}_t, P^1_t,\dots,P^K_t) - G_t(\overline P^p_t, \overline P^{op}_t, \overline P^1_t,\dots,\overline P^K_t)\\ &\qquad \quad + \overline G_t(\overline P^p_t, \overline P^{op}_t, \overline P^1_t,\dots,\overline P^K_t) - \overline G_t(P^p_t, \overline P^{op}_t, P^1_t,\dots, P^K_t))dt\geq 0, \end{aligned} $$

where $G_t$ is the function defined in the proof of Proposition 1, and $\overline G$ is the same function, but defined with measure flow $\bar m$ instead of m. Now, since $(P^p_t, P^{op}_t, P^1_t,\dots ,P^K_t)$ is the minimizer of $G_t$, $(P^p_t, \overline P^{op}_t, P^1_t,\dots , P^K_t)$ is the minimizer of $\overline G_t$ and the two functions are strictly convex, we deduce that the two price vectors are equal, possibly outside a set of zero measure.